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A CLASSIFICATION of CUBIC EDGE-TRANSITIVE GRAPHS of ORDER 18P

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A CLASSIFICATION of CUBIC EDGE-TRANSITIVE GRAPHS of ORDER 18P U.P.B. Sci. Bull., Series A, Vol. 77, Iss. 2, 2015 ISSN 1223-7027 A CLASSIFICATION OF CUBIC EDGE-TRANSITIVE GRAPHS OF ORDER 18p Mehdi ALAEIYAN1, M.K. HOSSEINIPOOR2 A graph is called edge-transitive if its automorphism group acts transitively on the set of its edge. In this paper, we classify all connected cubic edge-transitive graphs of order 18p for each prime p. Keywords: Edge-transitive graphs, Semisymmetric graphs, Symmetric graphs, s-Regular graphs, Regular coverings. 1. Introduction Throughout this paper graphs are assumed to be finite, simple, undirected and connected. For group theoretic concepts and notation not defined here, we refer the reader to [13, 24]. Given a positive integer n, we shall use the symbol to denote the ring of residues modulo as well as the cyclic group of order . For a graph X, we use V(X), E(X) and Aut(X) to we denote its vertex set, edge set and automorphism group, respectively. For , , is the edge incident to and in X and is the set of vertices adjacent to in X. For a subgroup N of Aut(X), denote by XN the quotient graph of X corresponding to the orbits of N, that is, the graph having the orbits of N as vertices with two orbits adjacent in XN whenever there is an edge between those orbits in X. A graph is called a covering of a graph with projection :→ if there is a surjection : → such that | is a bijection for any vertex and 1. A covering of with a projection is said to be regular (or k-covering) if there is a semiregular subgroup K of the automorphism group Aut() such that graph is isomorphic to the quotient graph K, say by h, and the quotient map → K is the composition h of and h. If is connected K becomes the covering transformation group. The fibre of an edge or vertex is its preimage under . An automorphism of is said to be fibre-preserving if it maps a fibre to a fibre, while every covering transformation maps a fibre onto itself. All of fibre-preserving automorphism from a group called the fibre-preserving group. Let G be a finite group and S a subset of G such that 1S and S = S-1 = {s−1|s S}. The Cayley graph Cay (G, S) on G with respect to S is defined to have 1 Prof., Department of Mathematics, Faculty of Science, South Tehran branch, Islamic Azad University, Tehran, Iran, e-mail: [email protected]. 2 Department of Mathematics, Iran University of Science and Technology, Iran 220 Mehdi Alaeiyan, M. K. Hosseinipoor vertex set G and edge set {gh| g,h G,gh−1 S}. A Cayley graph Cay (G, S) is connected if and only if S generates G. It is well known that Aut (Cay (G, S)) contains the right regular representation R(G) of G, the acting group of G by right multiplication, which is regular on vertices. A Cayley graph Cay (G, S) is said to be normal if R(G) is normal in Aut(Cay(G,S)). A graph is isomorphic to a Cayley graph on G if and only if Aut() has a subgroup isomorphic to G, acting regularly on vertices (see [4, Lemma 16.3]). Let s be a positive integer. An s-arc in a graph X is an ordered (s+1)-tuple ,,…,, of vertices of X such that 1 is adjacent to for 1 ≤ i ≤ s and 1 1 for 1 ≤ i ≤ s − 1. For a graph X and a subgroup G of Aut(X), X is said to be G-vertex-transitive, G-edge-transitive and G-s-arc-transitive if G acts transitively on the sets of vertices, edges and s-arcs of X respectively. It is easily seen that a graph X which is G-edge but not G-vertex-transitive is necessarily bipartite, with the two parts of bipartition coinciding with the orbits of G. In particular, if X is a regular graph, then these two parts have equal cardinalities, and such a graph is then referred to as being G-semisymmetric. In the case where G =Aut(X) the symbol G may be omitted from the definitions above, so that a graph X is called vertex-transitive, edge-transitive, s-arc-transitive and semisymmetric if it is Aut(X)-vertex-transitive, Aut(X)-edge-transitive, Aut(X)-s- arc-transitive and Aut(X)-semisymmetric, respectively. In particular 1-arc- transitive means arc-transitive or symmetric. A symmetric graph X is said to be s- regular if Aut(X) acts regularly on the set of s-arcs in X. Tutte [21, 23] showed that every cubic symmetric graph is s-regular for some 1≤ s ≤5. Tutte [22], proved that a vertex- and edge-transitive graph with odd valency must be symmetric. Thus a cubic edge-transitive graph is either symmetric or semisymmetric. The classification of cubic symmetric or semisymmetric graphs of different order is given in many papers. For example, the cubic symmetric graphs of order 4p, 4,6,6,10,10,16 and 16p2 were classified in [12,11,19,3] and the cubic semisymmetric graphs of order 2p3,6p2,6p3,8p and 8p2 were classified in [18,16,9,2,1]. In this paper, we obtain a classification of cubic edge-transitive graphs of order 18p. In order to state the main Theorem 1.1 we first introduce a family of cubic graphs. Let p be a prime such that p ≡ 1 (mod 3), and let k be an element of order 3 in . Set V (K3,3) = {a,b,c,x,y,z} to be the vertex set of the complete bipartite graph K3,3 with partite sets {a,b,c} and {x,y,z}. The graph CF18p is defined to have vertex set V (CF18p) = V (K3,3) × 3p and edge set E(CF18p) = {(a, i)(x, i), (a, i)(y, i), (a, i)(z, i), (b, i)(y, i), (b, i)(x, i + k + 1), (b, i)(z, i+1), (c, i)(x, i−1), (c, i)(y, i−k−1), (c, i)(z, i)|i 3p}. A classification of cubic edge-transitive graphs of order 18p 221 It will be shown in the Lemma 2.7 that the graph CF18p is independent of the choice of k and hence unique for given order. The following is the main result of this paper. Theorem 1.1 Let X be a connected cubic edge-transitive graph of order 18p, where p is a prime. Then X is either semisymmetric or s-regular for some s = 1, 2 or 5. Furthermore, (1) X is semisymmetric if and only if X is isomorphic to one of the graphs S54 and S126; (2) X is 1-regular if and only if X is isomorphic to the graph CF18p, where p ≡ 1 (mod 3); (3) X is 2-regular if and only if X is isomorphic to the graph F54; (4) X is 5-regular if and only if X is isomorphic to one of the graphs F90 and F234B. 2. Preliminaries Proposition 2.1 [18, Proposition 2.6] Let X be a G-semisymmetric graph for some subgroup G of Aut(X). Then either X ≡ K3,3 or G acts faithfully on each of bipartition sets of X. Proposition 2.2 [18, Proposition 2.3] Let X be a connected bipartite graph admitting an abelian subgroup G ≤ Aut(X) acting regularly on each of the bipartition sets. Then X is vertex-transitive. The next Proposition is a special case of [16, Lemma 3.2]. Proposition 2.3 Let X be a connected G-semisymmetric cubic graph with bipartition sets L(X) and R(X) and let N be a normal subgroup of G. If N is intransitive on bipartition sets, then N acts semiregularly on both L(X) and R(X), and X is an N-covering of a G/N-semisymmetric graph. Proposition 2.4 [15. Theorem 9] Let X be a connected symmetric graph of prime valency and G an s-arc-transitive subgroup of Aut(X) for some s ≥ 1. If a normal subgroup N of G has more than two orbits, then it is semiregular and G/N is an s-arc-transitive subgroup of Aut(XN), where XN is the quotient graph of X corresponding to the orbits of N. Furthermore, X is a regular covering of XN with the covering transformation group N. Let X = Cay (G, S) be a Cayley graph on a group G with respect to S. Set A :=Aut(X) and Aut(G,S) := {α Aut(G) | Sα = S}. Then we have: Proposition 2.5 [25, Proposition 1.5] The Cayley graph X = Cay (G,S) is normal if and only if A1 =Aut(G,S), where A1 is the stabilizer of the vertex 1 V (X) = G in A. Let p be a prime. It is easy to see that the equation x2 + x + 1 = 0 (1) has no solution in the ring 3 for p = 3. The following result determines the solutions of Eq. (1) in 3 for p ≠ 3. 222 Mehdi Alaeiyan, M. K. Hosseinipoor Lemma 2.6 Let p ≠ 3 be a prime. Then there exists k 3 solving Eq. (1) if and only if k is an element of order 3 in . 2 Proof. Suppose first that k 3p such that k + k + 1 = 0. Then k ≠ 1 and since k3 − 1 = (k − 1)(k2 + k + 1) = 0, it follows that k is an element of order 3 in .
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